Banknotes with a printed security image that can be detected with one-dimensional signal processing

ABSTRACT

A method to generate a security pattern to be embedded in an original image has the particularity that the detection of the security pattern can be achieved through a simple, low processing capability device. The method for generating a security image comprising the original image and the security pattern is characterized in that the security pattern is formed by an inverse transform of the combination in the frequency domain of an auxiliary image and a radially symmetric, two-dimensional pattern, said two-dimensional pattern created by sweeping a self-similar, one-dimensional function along a 360-degree arc, and the security image is generated by the modulation of at least one color of at least a part of the original image with the pattern.

STATE OF THE ART

Many solutions have been developed in the past to allow an easydetection of counterfeited documents. A different, and more directapproach, is to actually deter the counterfeiting operation. In thiscase, the document carries a security feature, which is detectable bythe hardware/software used for counterfeiting, and triggers an actionsuch as stopping the copying or scanning process. Existing solutions areeither based on optically visible features or invisible elements usingspecial consumables or digital signal processing methods. When focusingon features requiring no special consumables, such as security inks, thevisible solutions lack robustness against workarounds of counterfeitersand the invisible solutions put some constraints on the computationalpower and memory used by the detector. It should be noted that in bothcases feature detection is usually based and a digital image acquisitionfollowed by a signal processing method to digitally detect the securityfeature. As a consequence, detectors for invisible solutions cannot beimplemented directly into frequently used counterfeiting hardware havinglow computational capabilities (e.g. printers, scanners, monitors,digital cameras, etc.) but must be embedded in software at the computerlevel. The current invention describes a way to circumvent thislimitation by using a special detection/pattern combination allowing forboth visible and invisible features and low complexity detection.

Several techniques used for protecting valuable documents againstillegal duplication use small, localized variations of the visualappearance of the protected documents. These variations can take theform of a human-readable pattern (microtext, evolutionary screen dots[U.S. Pat. No. 6,198,545], moiré patterns [U.S. Pat. No. 5,995,638],microstructure color differences [EP 1073257A1]), or they can beimplemented using invisible, but machine-readable patterns (CryptoglyphWO01/00560, WO03/04178). In either case, authenticating a documentprotected by these methods requires the access to a significantly largedigitized area of the document at some or all times during theauthentication process. In digital signal processing this is translatedinto performing a computation on a 2D (two dimensional) matrix composedof pixel values of the acquired image.

This requirement poses two problems. A first problem arises with theauthentication of a document in the case where a minimum documentsurface is not available in its entirety at some time during theauthentication process. This is for instance the case for documents thatare digitally transmitted over a serial line or a bus system, e.g.document transmission from a scanner to a computer, from a camera to acomputer, from a computer to a printer, between two computers or betweena computer and a mobile phone.

A second problem arises when the authentication of documents has to beperformed by devices that have only little memory or a low processingpower. When the size of the document increases in a linear fashion, thememory and time required to process the document increase in ageometrical fashion. Therefore, authenticating security documents usedin everyday life, e.g. banknotes, plane tickets or ID cards, is a majorproblem for devices such as scanners, printers, digital cameras andmobile phones.

One important approach for invisible signal embedding is referred in theliterature as “digital watermarking”. Digimarc describes severalapproaches especially suitable for banknotes in U.S. Pat. No. 6,771,796,U.S. Pat. No. 6,754,377, U.S. Pat. No. 6,567,534, U.S. Pat. No.6,449,377. These approaches rely on modifications performed at amicroscopic level (i.e. 40 um or lower, corresponding to about 600 dpiresolution). These modifications are done in such a way that they can bedetected at a macroscopic level (i.e. using 100 dpi scanningresolution), but are generally invisible for the naked eye (Digimarcalso describes some techniques yielding to visible alterations in U.S.Pat. No. 6,674,886 and U.S. Pat. No. 6,345,104). The detection of thedigital watermark and decoding of the embedded data are performed usingcombinations of image processing algorithms which can be found in thedigital watermarking literature. Some of these algorithms include inparticular reference patterns in Fourier domain (for affine transformregistration), cross-correlation in the spatial domain (for registrationagainst image shift) and correlation in order to decode the signal. Itshould be highlighted that the most challenging part of the detectionprocess is usually to define a process that is robust againstgeometrical transformations as well as reaching satisfying reliabilityperformance. In some cases, a so-called “fragile digital watermarking”technique is used. With this technique, the embedded signal disappearswhen a copy of the protected document is performed. It enables todistinguish between original documents and copies. One example of suchan approach is described in WO2004/051917. Other approaches enable dataembedding in halftone images. Many solutions rely on an optical, analogprocess for revealing the data. However, some solutions are also basedon digital processing. In this case the common technique is to modifyslightly the threshold matrix in order to embed some information.Basically, any halftone image produced using this matrix and theoriginal gray level image carries the signal. One solution is describedin U.S. Pat. No. 6,760,464 (and U.S. Pat. No. 6,694,041) and anotherapproach is also presented in U.S. Pat. No. 6,723,121 each with adifferent watermarking technique. A more generic approach which does notspecify a particular digital watermarking technique is described in U.S.Pat. No. 6,775,394. Some approaches do not use digital watermarkingtechnique (in the sense of robust steganography), like in U.S. Pat. No.6,839,450 where authors describe a detection method of data embedded inhalftone images using matched filter. It is possible to significantlyimprove embedding performance in half-tone images by using modifiedversion of more sophisticated half-toning scheme. For instance,US2003021437 gives a description of a generation of a dither matrixproduced from a bitmap using morphological operations. Dither matrix isthen used for producing halftone images, which may be used in securityprinting. Inserting a signal into a digital media or printing it on adocument and detecting it later has been address extensively in olderpatents. From a technical point of view the main issues to solve aresignal design, signal embedding and signal detection. Here, signal canbe a modification applied to an existing image or the generation of anindependent signal and printing it over an existing document overlayingit onto a digital image. Signal design is largely driven by the functionbehavior of the detector. It is desirable that the detector can detector retrieve the embedded signal independently of possible geometricaltransformations applied to the protected media. To solve this challengeit is state of the art in digital marking technologies to either embedadditional key characteristics in the spatial or even frequency domainthat later allow for the identification of the geometricaltransformation and its inversion (for instance the U.S. Pat. No.6,408,082, U.S. Pat. No. 6,704,869 and U.S. Pat. No. 6,424,725 describeapproaches where a log-polar in the transform domain is used to computethe geometrical transform). A different approach is based on the designand embedding of an auto-similar signal. During detection anauto-correlation function is computed. The analysis of theauto-correlation function then allows for the identification of thegeometrical transformations and their inversions.

All the above solutions solve the problem of robust detection using2-dimensional processing techniques for continuous or halftone images.However, none of them perform this detection using a 1D signalprocessing, which is required for applications based on low computingpower systems.

A 1D solution is described in AU 2002951815 where the inventors proposedan approach to mark digital images with embedded signal where thesignals are represented by a 2D pattern constructed using a 1D basisfunction. For the detection of the pattern, the inventors first computea projective transformation of the image in and then retrieve theembedded information through a 1D correlation at different angles.However, since the correlation has to be re-computed for each angle, theoverall complexity is still of the same order as for the 2D processingdescribed above.

BRIEF DESCRIPTION OF THE INVENTION

The present invention aims to propose a method to generate a securitypattern comprising a printed security image, said image comprising anoriginal image and a security pattern, characterized in that, saidsecurity pattern being obtained by a predefined inverse integraltransform of the combination between an auxiliary image and atwo-dimensional pattern created by sweeping a one-dimensional functionalong a predefined curve, such as said security pattern being detectablefrom correlation properties of one line of said secured banknote sampledalong any arbitrary direction and with any resolution between 50 and1200 dots per inch, the security image being generated by the merging ofat least one color of at least a part of the original image with thesecurity pattern.

The present invention consists of two methods summarized below:

-   -   The first method is used for generating a security document by        applying a security pattern to an original document, for        instance under the form of a linear grating. This method for        generating a security image, such image comprising an original        image and a security pattern, has the particularity that the        security pattern has the form of a signal swept along a        predefined curve, in which the lines width and/or the line        spacing is modulated to embody a predefined data, the security        image is then generated by the modulation of at least one color        of at least a part of the original image with the grating.        Basically, in the particular case where the curve is a straight        line used in the spatial domain, this security pattern is        similar to a barcode. In another particular case, the curve        takes the form of a circle in the domain of a predefined        integral transform (e.g. a Fourier transform or a Hilbert        transform), and the pattern is combined with an auxiliary image        before the combination undergoes an inverse integral transform.        The result of this inverse integral transform is then merged        with the original image using an approach based on digital        halftoning. Some of the underlying principles of another merging        process are defined in AlpVision patent CH694233. This approach        is based on the overprint of a low density set of dots over an        image. Doing so creates a so-called “asymmetrical modulation”        (since generally printing inks only decrease the local        luminance) which is used to secretly embed a signal.    -   The second method is used for authenticating a security document        generated with the first method by detecting the presence of the        security pattern in arbitrarily located, rotated and scaled        lines of the document (the independence from scaling factor        enables to successfully detect the security pattern on a whole        range of printing resolutions, typically from 50 to 1200 dpi).        The detection is performed using a one-dimensional signal        processing. This enables a very low complexity computation        compared to classical image processing approaches described in        the state of the art above. In particular, it is then possible        to embed the detection process into simple hardware like printer        or scanner, enabling to implement a functionality of counterfeit        deterrence by stopping the copy process when a banknote is        detected.

BRIEF DESCRIPTION OF THE FIGURES

The invention will be better understood thanks to the attached Figuresin which:

FIG. 1 shows a linear grating of alternating stripes.

FIG. 2 shows a grating including a square-pulse signal.

FIG. 3 shows an example of stripe spacing modulation.

FIG. 4 shows an example of stripe width modulation.

FIG. 5 shows the steps to a linear interpolation of an image.

FIG. 6 shows a first example in which the stripes varies continuouslyalong their width.

FIG. 7 shows another example type of stripe's modulation.

FIG. 8 to FIG. 11 show various example of stripes modulation.

FIG. 12 shows an example of a pattern exhibiting an invariant feature.

FIG. 13 shows an example in which the image is split into sub-areas,each of them embedded with a different security pattern.

FIG. 14 shows an example, in which the pattern is embodied in each colorcomponent.

FIG. 15 shows various pattern intensities.

FIG. 16 shows two magnifications of a halftone image produced with apattern exhibiting an invariant feature.

FIG. 17 shows a thickening modulation of a text.

FIG. 18 shows the signals for the referenced signal and the signal afterform a rotation of the image.

FIG. 19 shows the logarithm value of the signal stretched.

FIG. 20: A general iterative detection scheme for each line of thebanknote.

FIG. 21 shows an auto-correlated, one-dimensional signal built bysumming a set of periodical functions that differ only by their period.

FIG. 22 shows a circularly symmetric, two-dimensional signal built bysweeping an auto-correlated, one-dimensional signal.

FIG. 23 shows a self-similar, one-dimensional signal built byrecursively replacing portions of a simple function with scaled-downcopies of itself.

FIG. 24 shows a circularly symmetric, two-dimensional signal built bysweeping a self-similar, one-dimensional signal.

FIG. 25 shows a one-dimensional signal that is scale-invariant across agiven range of scaling factors.

FIG. 26 shows a circularly symmetric, two-dimensional signal built bysweeping a scale-invariant, one-dimensional signal.

FIG. 27 shows another one-dimensional signal that is scale-invariantacross a given range of scaling factors.

FIG. 28 shows another circularly symmetric, two-dimensional signal builtby sweeping a scale-invariant, one-dimensional signal.

FIG. 29 shows a one-dimensional, bandpass filter built by a combinationof Butterworth filters.

FIG. 30 shows a two-dimensional, bandpass filter built by sweeping aone-dimensional, bandpass filter.

FIG. 31 shows two superimposed representations of the same dithermatrix.

FIG. 32 shows a three-dimensional representation of a spot function.

FIG. 33 shows an example of a dither matrix and a bilevel halftonegradation obtained with this dither matrix.

FIG. 34 shows a large dither matrix built by tiling the plane withmultiple copies of a smaller dither matrix.

FIG. 35 shows the embedding of a circularly symmetric pattern in thefrequency domain and a bilevel halftone gradation.

FIG. 36 shows the result of various morphological operations applied toa discretized spot function.

FIG. 37 shows the module of the Fourier transform of morphologicaloperations applied to an embedded pattern.

FIG. 38 shows the construction of a dither matrix based on the resultsof various morphological operations.

FIG. 39 shows a bilevel halftone gradation produced by thresholding agrayscale image with a morphological dither matrix.

FIG. 40 shows the combination of a spot function and a circularlysymmetric pattern in the frequency domain.

FIG. 41 shows a bilevel halftone image generated with a spot functionbased on a balanced circularly symmetric pattern.

FIG. 42 shows a bilevel halftone image generated with a spot functionbased on a combination of two balanced circularly symmetric patterns.

FIG. 43 shows how patterns can be combined in the same banknote.

FIG. 44 shows the building of a security image layer by the shifting ofthe security pattern.

FIG. 45 shows a radially symmetric pattern with a coarse-grain randomradial jitter.

FIG. 46 shows a radially symmetric pattern with a fine-grain randomradial jitter.

FIG. 47 shows a radially symmetric pattern generated with a function.

FIG. 48 shows another radially symmetric pattern generated with afunction.

FIG. 49 illustrate an interactive or automatic process for signalintegration into the design art work.

FIG. 50 shows the results of the normalized cross-correlations between atemplate and two signals.

FIG. 51 shows a general diagram of the invention.

FIG. 52 shows the projections of a radially symmetric pattern before andafter a rotation.

DETAILED DESCRIPTION OF THE INVENTION

Signal Embedding

The signal is embedded by overprinting a light and visuallynon-disturbing pattern across an existing design (the pattern can beoverlaid in digital domain). The visual disturbance induced by theembedded pattern is kept below the visual perception threshold thanks toa combination of two factors. First, the chromatic variations induced bythe embedded pattern are kept under a specific visual threshold based onjust noticeable differences (Melgosa, M., Hita, E., Poza, A. J., Alman,David H., Berns, Roy S., Suprathreshold Color-Difference Ellipsoids forSurface Colors, Color Research and Application 22, 148-155, June 1997.).Secondly, the spatial frequency of the pattern is kept at sufficientlyhigh value, so that the chromatic contrast formed by its individualparts goes unnoticed (McCourt, Marc E., Spatial frequency tuning,contrast tuning, and spatial summation of suprathreshold lateral spatialinteractions: Grating induction and contrast-contrast, OSA AnnualMeeting Technical Digest 16, 155, 1993). The joint use of thesechromatic and frequency criteria enable to obtain simultaneously asecurity pattern that combines the advantages of a low resolution(compared to the resolution of existing design), a high signal amplitudeand a low visibility (as shown in (FIG. 13 and FIG. 14).

A second method for embedding the signal uses the linear grating as abasis for producing a halftone screen. With this method, the width ofthe stripes composing the grating varies accordingly to the intensitylevels present in the original image (see FIG. 15). A security documentgenerated by such a method takes the form of a halftone image renderedwith a line-based halftone screen (see FIG. 16).

A third method for embedding the signal in printed images uses aprinting process capable of producing stripes with a controllablethickness, such as intaglio printing. With this method, the securitypattern is printed as an overlay on the original image, either by usingan additional intaglio plate or by modifying an already existing plate.By using a transparent or a semi-transparent ink (e.g. a varnish) and bycontrolling the thickness of the printed stripes, it is possible tocontrol the embedding strength of the overlaid pattern.

A fourth method for embedding the signal in microstructure, digitalimages (e.g. halftone images or digital images containing a microtext)consists in applying local modifications to the microstructure. Theselocal modifications have the effect of thickening the microstructure inthe parts where the stripes of the pattern are thicker, and they havethe effect of thinning the microstructure in the parts where the stripesof the pattern are thinner (FIG. 17). At a macroscopic level, areas witha thickened microstructure have a higher intensity value and areas witha thinned microstructure have lower intensity.

A fifth method replaces the linear grating image by a circularlysymmetric grating image. This circularly symmetric grating is obtainedby sweeping a one-dimensional signal across a 360-degree arc. Theproperty of circular symmetry guarantees that the signal observed alonga straight line crossing the grating at its center remains the same forall angles of the line. The signal is then embedded using the first, thethird or the fourth method. Examples of circularly symmetric signals areprovided in FIG. 22, FIG. 24, FIG. 26 and FIG. 28. In FIG. 22, the 2Dsignal is built by sweeping the auto-correlated 1D signal depicted inFIG. 21 along a 360-degree arc. In FIG. 24, the 2D signal is built bysweeping the self-similar 1D signal depicted in FIG. 23 along a360-degree arc. In FIG. 26, the 2D signal is built by sweeping thescale-invariant 1D signal depicted in FIG. 25 along a 360-degree arc. InFIG. 28, the 2D signal is built by sweeping the scale-invariant 1Dsignal depicted in FIG. 27 along a 360-degree arc.

A sixth method for embedding a circularly symmetric grating uses aninverse integral transform. An integral transform is any transform T_(f)of the form:

T_(f) = T(f(u)) = ∫_(t 1)^(t 2)f(t)K(t, u) 𝕕t

where the function K(t,u) is the kernel of the transform. The simplestexample of an integral transform is the identity transform, withK(u,t)=δ(u−t) (δ is the Dirac distribution), t₁<u, t₂>u. Another exampleis the Laplace transform, with K(u,t)=e^(−ut), t1=0, t2=∞. Yet anotherexample commonly used in signal processing is the Fourier transform,with

${{K\left( {u,t} \right)} = \frac{{\mathbb{e}}^{{\mathbb{i}}\; u\; t}}{\sqrt{2\;\pi}}},{{t\; 1} = {- \infty}},{{t\; 2} = {\infty.}}$

The selected inverse integral transform is applied to a pair ofcomponents. The first component is a module component R; it is generatedwith a circularly symmetric grating. The second component is a phasecomponent P; it is generated with the output of a quantum random numbergenerator (e.g. http://www.randomnumbers.info/) or a pseudo-randomnumber generator. The module component are used together to produce anarray A of complex numbers using the relationC(x,y)=R(x,y)*exp(i*P(x,y)), where i denotes the square root of −1. Theresult A* of the inverse Fourier transform of C yields a signal thatlooks like white noise, but that exhibit the original grating in thefrequency domain. The signal A* is then printed onto the banknote usingthe first, the third or the fourth method. FIG. 35 shows an example ofembedding a circularly symmetric grating in the frequency domain. AFourier transform (H) is synthesized by combining a module based on acircularly symmetric signal (1201) and a phase based on white noise(1202). The inverse Fourier transform of (H) yields a two-dimensionalsignal (1203) that looks like white noise.

A seventh method uses a circularly symmetric grating embedded in thefrequency domain as a spot function for thresholding a grayscale image.An example of a three-dimensional representation of a general spotfunction is shown in FIG. 32: the values of the spot function arematerialized by steps of varying heights that have a grayscale valuecorresponding to their height. The embedded spot function is thendiscretized in so as to produce a dither matrix that can be used tothreshold a grayscale image in order to generate a bilevel halftoneimage. An example of a dither matrix is shown in FIG. 31: a firstrepresentation is given by an array of numerical thresholds that areuniformly distributed between 0 and 255, and a second representation ofthe same dither matrix is given by an array of grayscale values thatcorrespond to the numerical thresholds of the first representation. FIG.33 shows another example of a dither matrix (901) represented as anarray of grayscale values; this dither matrix is used to threshold alinear grayscale gradation in order to produce a bilevel halftonegradation (902). The size of the dither matrix may be adapted to thesize of the circularly symmetric pattern by building a second, largerdither matrix as a tiling of the first dither matrix, as shown in FIG.34. By construction, a halftone image thresholded using a dither matrixbuilt with an embedded spot function will exhibit the embeddedcircularly symmetric grating in the frequency domain. Thistwo-dimensional signal is normalized so as to yield the desired spotfunction. FIG. 35 shows an example of using a two-dimensional signal(1203) as a spot function in order to threshold a linear grayscalegradation in order to produce a bilevel halftone gradation (1204).

An eighth method builds an embedded spot function based on a signal A*constructed with the fifth method. The continuous signal A* isthresholded in order to produce an array B of black and white pixels.The array B is duplicated so as to produce identical copies {B₁, B₂, . .. B_(n)}. Each copy B_(k) (k=1 . . . n) undergoes a different series ofmorphological operation such as inversion, dilation, erosion, pruning,opening, closing, skeletonization, extraction of outlines. FIG. 36 showsan example of morphological operations applied to a discretized spotfunction. A square area (601) of the spot function (1203) depicted inFIG. 35 is thresholded (602) so that half its elements are black and theother half are white. The outlines of this bitmap are shown in (604).The skeleton of the same bitmap is shown in (606). The pruned skeletonof the same bitmap is shown in (608). The values of the thresholdedbitmap are inverted so as to produce a dual bitmap (603). The inverseoutlines of this dual bitmap are shown in (605). The inverse skeleton ofthe same dual bitmap is shown in (607). The inverse pruned skeleton ofthe same dual bitmap is shown in (609). By construction, the results{M₁, M₂, . . . M_(n)} of the morphological operations will all exhibitto some degree the embedded circularly symmetric pattern in thefrequency domain. This property is illustrated by FIG. 37, which showsthe module of the Fourier transform of some of the morphological resultsdepicted in FIG. 36. The same circularly symmetric pattern is visiblewith a variable extent and a variable clarity in each one of thetransforms (1202), (1204), (1206) and (1208). The results {M₁, M₂, . . .M_(n)} of the morphological operations are then measured: for each M_(k)(k=1 . . . n), the ratio K_(k)/N_(k) is calculated, where K_(k) is thenumber of black pixels in M_(k) and N_(k) is the total number of pixelsin M_(k). The results of the morphological operations {M₁, M₂, . . .M_(n)} are ranked according to their ratio of black pixels K_(k)/N_(k).For each M_(k), the black pixels are replaced by the value K_(k)/N_(k).In the final step, all the M_(k) are merged together to form a spotfunction S. The values of the individual pixels of S are calculatedusing the relation: S(x,y)=max_(k)(M_(k)(x,y)). The ranking of themorphological steps (702-708) and their merging into a dither matrix(709) is illustrated in FIG. 38. After the merging, the dither matrixcan be further enhanced in order to obtain an equilibrated dithermatrix. Such an enhancement can take the form of weighted histogramequalization, or a slight Gaussian blur, or the addition of a smallamount of noise. In FIG. 39, a dither matrix based on morphologicaloperations is used to threshold a linear grayscale gradation in order toproduce a bilevel halftone gradation.

A ninth method builds an embedded spot function by combining a generalspot function and a circularly symmetric pattern in the frequencydomain. FIG. 40 shows the construction of such a combined spot function.The general spot function is embodied by the tiling (1001) of multiplecopies of a simple spot function traditionally used to generateclustered-dot, amplitude-modulation halftone screens (1002). This tilingis transposed to the frequency domain by the means of a Fouriertransform (F), and the result of this Fourier transform is thendecomposed into a module component (1003) and a phase component (1004).A circularly symmetric pattern (1005) is combined with the modulecomponent by the means of a linear interpolation (I). Other possiblecombination schemes can be used, such as a multiplicative scheme, aquadratic scheme or an exponential scheme. The combined module component(1006) is merged back with the phase component (1004) using an inverseFourier transform (H). The result of this inverse Fourier transformundergoes a histogram equalization so as to produce a balanced spotfunction (1007). As an example, this spot function is used to thresholda grayscale patch of constant value in order to produce a bilevelhalftone patch (1008).

The above methods are not limited to a circularly symmetric grating ofFourier module; it can also be applied with any pattern obtained bysweeping a particular 1D signal in an integral transform domain.

A tenth method produces a dither matrix in the spatial domain by using abalanced, circularly symmetric pattern as a spot function. FIG. 41illustrates this method with a bilevel halftone image generated by usinga LRHF as spot function in order to threshold a linear grayscalegradation.

An eleventh method combines two or more spot functions generated withthe tenth method in order to produce a new spot function. Combinationschemes include arithmetic operations such as addition, subtraction andmultiplication, N-cyclical group operations such as addition modulo N,subtraction modulo N and multiplication modulo N, geometric operationssuch as translation, scaling and rotation, and logical operations suchas OR, AND and XOR.

FIG. 42 illustrates this method with a bilevel halftone image generatedby using a spot function based on the combination of two circularlysymmetric patterns. The patterns used in this example are a LRHF and atranslation of the same LRHF. The combination scheme used is an additionmodulo 256.

Signal Detection

The embedded pattern is typically recovered after its print-out. Adigital imaging device (like a digital scanner or a camera for instance)is then used to bring back the printed material in the digital domain.The pattern is designed in such a way that it is possible to triggerdetection with a mono-dimensional signal processing performed along astraight line having an arbitrary direction across the pattern, for anyscale and rotation transformations (in a previously defined range). Twoissues have to be addressed in order to obtain this result: thereliability of the detection trigger (false-positive and false-negativedetections) and the robustness to geometrical transforms.

The reliability of the detection basically relies on a statistical test.This test must be performed on a sufficiently large set of data in orderto reach the desired false-positive (signal detected while not beingpresent) and false-negative performance (signal not detected while beingpresent). In the targeted application, the false-positive rate isexpected to reach 1 over 10 millions or better. The statistical data canbe processed during the digitization or during the printing process.Since the detection approach relies on a 1 dimensional signalprocessing, it may also be performed in real-time as data is streamedinto the hardware into which the detection is performed.

The robustness to geometrical transforms can be achieved using twodifferent approaches. One solution is to have a signal that is invariantwith affine transformations; the other solution is to compensate for thetransformation before decoding the signal.

Invariant Signal Approach

The pattern is designed so that the 1D profile of the pattern taken inany direction and with any scale, exhibits an invariant feature. Thissimilar feature can then be used to trigger the detection, disregardingthe geometrical transform which has been applied to the image. FIG. 12shows an example of a pattern exhibiting an invariant feature: thispattern is composed of concentric circles. Any straight line crossingthis pattern through its center will produce the same 1D profile. FIG.16 shows a pattern exhibiting an invariant feature embedded into animage under the form of a halftone screen composed of concentriccircles.

Invariance under rotation can also be obtained by embedding a circularlysymmetric pattern in the Fourier domain. When an image is processed by aprinting device or an acquisition device, image data is transferredthrough the device one line at a time. The detector applies a colortransform to the individual image lines in order to transpose them intothe color space where the security image is present. The sum S of thetransformed lines is stored in a separate image buffer. This sum can beviewed as the projection of the image from a two-dimensional space ontoa one-dimensional space. After a predefined number of lines have beensummed, the detector calculates the one-dimensional Fourier transform FSof the sum S. The result of this Fourier transform is individuallycompared to a bank of predetermined one-dimensional signal templatesstored in the device's ROM. These comparison operations belong to theclass of matched filtering, and they are implemented with across-correlation (normalized cross-correlation, phase-only crosscorrelation, canonical cross-correlation). This process is illustratedin FIG. 50, which shows the result (1303) of a normalizedcross-correlation between a scale-invariant template signal (1301) and amirror copy of the same signal (1302). As a comparison, the result(1305) of a cross-correlation between the same template signal (1301)and white noise (1304) is shown. Before the comparison takes place, FScan undergo a series of pre-processing steps in order to increase thereliability of the cross-correlation. These steps include windowing(Hamming), pre-whitening, band-pass filtering, histogram equalization,envelope demodulation, denoising, windowed averaging. The result of thecomparison between FS and the device's bank of one-dimensional signaltemplates is assessed with the help of one or more statistical tests. Ifthe assessment yields a positive answer, the image is assumed to carrythe security image and the device reacts accordingly by interrupting itsfunction. This process may also be performed in several steps: a firststep using a few lines to detect if the signal is present. If the signalis detected then additional lines are processed in order to confirm thedetection (this approach enables to satisfy false-positive requirementsand processing speed requirements). Data of successive lines may also beused to compute a signal in a rotated direction. This also contributesto reach a desired false-positive detection rate.

The condition of circular symmetry is necessary to guarantee a strictinvariance under rotation, but such a strict invariance is not alwaysneeded in order to get a two-dimensional pattern that can be reliablydetected in one dimension. Two-dimensional signals that observe the lessstrict requirement of radial symmetry can also be detected reliably inone dimension if they are based on a one-dimensional signal that iseither autocorrelated, self-similar or scale-invariant (or has severalof these properties). FIG. 45 shows such a radially symmetric patterngenerated by subdividing a scale-invariant pattern (LRHF) in 36 sectorsof 10 degrees of arc and by applying a random radial jitter to eachsector. FIG. 46 shows another radially symmetric pattern generated bysubdividing a scale-invariant pattern (LRHF) in 360 sectors of 1 degreeof arc and by applying a random radial jitter to each sector. FIG. 47shows a radially symmetric pattern generated with a function of theform:F(R,Theta)=cos(a*log 2(R)+b*max(0, cos(k*Theta)))

FIG. 48 shows another radially symmetric pattern generated with afunction of the form:F(R,Theta)=cos(a*log 2(R)+b*abs(cos(k*Theta)))

Because radially symmetric patterns above are based on a scale-invariantfunction, the sum of their lines will produce a one-dimensional signalwith a shape that remains similar when the patterns are rotated. Thisproperty means that the cross-correlation between a one-dimensionalsignal template and the projection of such a radially symmetric patternwill produce a similar response regardless of the orientation of thepattern. FIG. 52 provides an illustration of this property.

Compensation Based Approach

Compensation can be performed either by using a separate referencepattern (for instance a printed circular pattern enables to define thehorizontal versus vertical scale alteration) or by a mathematicaltransform of the signal that maps it into another domain in which thecompensation is performed more easily. For instance, a logarithmictransform enables to map the signal in a different space that allows foreasy compensation of a scale alteration. This scaling can be caused forinstance by a digitizing resolution that is different from the printingresolution of the signal. It may also be caused by a rotation of thedigitized sample as shown in FIG. 18. The scaling factor is related tothe angle of rotation a with the cosine function Cos(α).

Indeed, let s(x)=f(log(x))

If the original signal o(x) differs from s(x) by a factor λ (See FIG.19), then:s(x)=o(λx)Using the log transform gives:s(Ln(x))=o(Ln(λx))Then it follows that:s(t)=o(Ln(x)+Ln(λ))=o(t+Δt),with t=Ln(x) and λ=exp(Δt)

This equation means that the stretched signal s(x) is equivalent to atranslation when a log scale is used to define the sampling position asshown in FIG. 19. The value of this translation can be found by usingthe maximum value of the cross-correlation signal computed between thedigitized signal f(x) and the known original signal o(x). It thenenables to compute the scale factor using the equation:and λ=exp(Δt)

It is then possible to retrieve the angle α from λ and compensate forthe rotation by a rotation with reverse angle.

Preferred Embodiments for Pattern Detection

The statistical test is performed in the simplest embodiment as a finitestate machine which counts how many times the signal matches somepredefined characteristics and compares it to a threshold. Thesecharacteristics can be a number of transitions of the signal, a sequenceof width as shown in FIG. 4 or a sequence of spacing as shown in FIG. 3.The signal is then defined as a grayscale value. In another embodiment,the signal is a vector defined by several color components, for instancered-green-blue, cyan-magenta-yellow-black, hue-luminance-saturation,hue-saturation-value, CIE-Lab, CIE-Lch or CIE-XYZ (or in certainpredefined range of light wavelength). This multi-color approach enablesto increase the detection rate performances. In another embodiment thedetected characteristics are defined by a quantum random numbergenerator or a pseudo-random number generator with a key providedseparately or computed from other features (visual or not) of thesecurity document.

In another embodiment the statistical test is performed using signalprocessing algorithms (for instance but not limited tocross-correlation, invariant computation, etc). The result of this testis then compared to some pre-defined threshold or threshold computedfrom the processed data.

The robustness to geometrical attacks can be performed in one embodimentby the mean of an invariant feature, including but not limited to,circular patterns. In another embodiment, the robustness is obtainedusing a compensation method. In one embodiment this method uses theabove described Log transform combined with some cross-correlation (orother matching indicator) technique. The general detection scheme isshown in FIG. 20: In 2600, colors of the banknote are digitally sampledalong a straight line across the banknote (and titled with an arbitraryangle) and stored as a 1D signal. In 2601, a filtering may be performedin order to enhance some particular properties. In 2602, a statisticaltest is then performed. This test can be based for instance on thecross-correlation with a 1D signal, or an autocorrelation, a measurementof auto-similarities, etc. Such measurements are generically named“correlation” throughout this document. In 2603, this valuescorresponding to this measurement is accumulated with values computedfor previous lines and compared to one or several thresholds. If theaccumulated values exceed some threshold, a positive detection signal issent in 2604. In case of no positive detection, the system acquires anew line of the banknote in 2605. The detection of the security imagemay also use one-dimensional signal processing based on Fouriertransform. Its theoretical basis lies on a result from the field oftomographic reconstruction, the projection-slice theorem. This theoremstates that the Fourier transform of the projection of a two-dimensionalfunction onto a line is equal to a slice through the origin of thetwo-dimensional Fourier transform of that function which is parallel tothe projection line. The corresponding detection scheme is still shownin FIG. 20 with the addition of a Fourier transform in 2601.

Preferred Embodiments for Pattern Creation

In its simplest embodiment, the security pattern that is applied by thefirst method takes the form of a linear grating of alternating dark andlight stripes (FIG. 1). This grating incorporates a square-pulse signal(FIG. 2) which is carried by the modulation of the distance between thecenters of the stripes (FIG. 3) or by the modulation of the width of thestripes (FIG. 4).

The security document is obtained by embedding the security pattern intothe original image by the means of a linear interpolation. If C(x,y) isthe value of the original image at the position (x,y), P(x,y) is thevalue of the pattern at the position (x,y) and W(x,y) is the desiredweight of the pattern at the position (x,y), then the value S(x,y) ofthe security document at the position (x,y) is calculated with:S(x,y)=(1−W(x,y))*C(x,y)+W(x,y)*P(x,y)

By the appropriate choice of W(x,y), it is possible to continuously varythe visibility of the pattern from totally invisible to totally visible.

In a second embodiment of the invention, the value of the stripes variescontinuously along their width. With this variation, the shape of thesignal carried by the security pattern takes the form of a continuousfunction like a sine wave (FIG. 6) or a triangle pulse (FIG. 7).

In a third embodiment of the invention, the pattern undergoes ageometrical transform under the form of a conformal mapping (FIG. 8,FIG. 9, FIG. 10, FIG. 11). A particular case of a geometrical transformproduces a pattern formed of concentric circles (FIG. 12). Such apattern exhibits an invariant feature: the same signal can be detectedacross all the straight lines crossing the pattern through its center,regardless of their orientation. Such an invariant feature enables thedetection approach based on an invariant signal.

Someone skilled in the art will also be able to realize aboveembodiments with any pattern obtained by sweeping a constant or varyingsignal.

In a fourth embodiment of the invention illustrated in the FIG. 13, theoriginal image is divided in several separate areas and the securitydocument is obtained by embedding a different security pattern in eacharea.

In a fifth embodiment of the invention, the security document isobtained by separately embedding a different security pattern into ineach color component of the original color image (FIG. 14). (RGB imagesembedded in the B component, CIE-Lab images embedded in the L component,CMYK images embedded in the Y component, etc.)

In a sixth embodiment of this invention, the security document isobtained by transforming the color space of the original image beforeembedding the pattern into a subset of the transformed color components.(RGB->HLS, embedding in the H component; RGB->CIE-Lch, embedding in thec component; etc.)

In a seventh embodiment, the security pattern is embedded in thesecurity document by modifying only the chrominance components of theoriginal image. The original luminance component is left unmodified, andthe difference between the original chrominance components and themodified chrominance components is maintained below the perceptualthreshold.

In an eight embodiment, one security pattern is generated for everyluminance level present in the original image. The thickness of thelines of these patterns varies accordingly to the luminance level theyare associated to, but the position of these lines remains constantacross each one of the patterns (FIG. 15). The security document is thenobtained from these security patterns by embedding them under the formof a halftone screen (FIG. 16). Using a circular pattern (like theexample shown in FIG. 16) enables to obtain a signal that is invariantto rotation.

In a ninth embodiment, the security pattern is totally visible (W(x,y)=1in the previous equation for (x,y) belonging to marked area) on selectedareas of the document.

In a tenth embodiment, the security pattern is an invariant signal whichis defined in the Fourier domain. A security image layer is built fromthe security pattern by the means of an inverse Fourier transform.

-   1. Security image layers all have these common properties:    -   1.1. Layer is printed onto a banknote.    -   1.2. Layer is bi-level (ink/no ink).    -   1.3. Layer is generated by applying a dither matrix to a        grayscale image in order to obtain a halftone.        -   1.3.1. Layer produces a visible pattern in frequency domain            with a circular symmetry or a central symmetry. Pattern is            built by applying a 360-degree circular sweep to a            one-dimensional signal. This one-dimensional signal has at            least one of the three following properties:            -   1.3.1.1. one-dimensional signal is self-similar across a                given range of scale factors (e.g. a fractal signal).            -   1.3.1.2. one-dimensional signal is auto-correlated                across a given range of scale factors (e.g. a                Cryptoglyph).            -   1.3.1.3. one-dimensional signal is invariant across a                given range of scale factors (e.g. a log-harmonic                function).

Basically any two-dimensional function f depending on the radius r andthe angle theta is possible as long as f(r,theta)=f(r, theta+pi) andf(r) is self-similar, auto-correlated or scale invariant.

When the signal is invariant across a given range of scale factors(typically for a log constructed signal), it is possible to shiftarbitrarily (for instance using a quantum random number generator or apseudo-random number generator) the signal along the radius fordifferent angles. Let us consider the particular case of the functionbelow:f(r,θ)=Cos(a Ln(r)+kθ+φ)In this equation, k and a are two fixed parameters. Then, φ is the shiftof the signal.

FIG. 44 illustrates this process in the Fourier space 903. The periodicsignal in sector 901 and 902 only differ by their phase. The sectors 904and 905 are symmetric versions of respectively sectors 902 and 901. Inthese cases, the phase φ is actually a function of the angle theta andof the radius r. The approach enables to better conceal the signal inthe Fourier domain and thus making it more difficult for an attacker todetect and remove it. It also enables to strengthen the signal for somesets of angle and radius values, which may be useful to increase thedetectability of the signal (for instance if the banknote artworkfrequencies interfere with the signal in the Fourier domain or toenhance the detectability at 0 and 90 degrees in the Fourier domain).Other examples are shown in FIG. 45, FIG. 46, FIG. 47 and FIG. 48 withdifferent φ functions (where φ is a random function in FIG. 45 and FIG.46).

-   2. Dither matrices are created with the use of one or more spot    functions.-   3. A first class of spot functions is based on a pair of 2D    matrices. The first matrix (A) contains a visible pattern according    to 1.3.1; the second matrix (B) contains additive white noise (but    any other type of noise can also be used) in the range [−pi, pi] in    order to obtain a rather uniform image in the spatial domain. These    two matrices are converted to a single matrix of complex numbers    (C), with C(x,y)=A(x,y)*exp(i*B(x,y)). C is then made symmetrical    (FFT sense), so that its inverse Fourier transform is a real image.    The spot function used for generating the security image is obtained    by calculating the inverse Fourier transform of C. It is also    possible to use an centrally asymmetrical C matrix. In this case,    the inverse Fourier transform is a complex image. Real and imaginary    parts can be printed with different colors, so that the detector can    recover the complex image. Not only colors can be used in order to    aid the decoder to distinguish between the real and imaginary parts.    It possible to use any optical property which provides two    independent channels for the real and imaginary parts. For instance,    the top half portion of a banknote area may encode the real part    while the bottom part will encode the imaginary part. Any other    spatial criteria known by the decoder may be used to differentiate    areas dedicated to real and imaginary parts (like real part always    encoded in circular areas or borders of the banknote, etc. . . . ).    Another way to construct the security image defined as    A(x,y)*exp(i*B(x,y)) is to use a matrix A(x,y) with one of the above    method and a phase matrix B(x,y) which coefficients are not all    randomly chosen (the FIG. 35 illustrates the way the security image    is designed for the particular case of a totally random phase matrix    1202). In this case, we have:    B(x,y)=r(x,y) for (x,y) belonging to S1    B(x,y)=f(x,y) for (x,y) belonging to S2

Where r(x,y) is a quantum random number or a pseudo-random numberbetween [−pi,pi] and f(x,y) is an arbitrary function with values between[−pi,pi], S1 and S2 are two sets of (x,y) indexes such that S1∪S2 is thewhole image.

For instance, low frequencies may be random while high frequencies maybe fixed with a constant value. In this case, the corresponding inverseFourier transform of A(x,y)*exp(i*B(x,y)) will not be a uniform noise.One interest of this approach is to create a decorative pattern in thespatial domain.

-   4. A second class of spot functions is obtained by combining a spot    function F1 of the first class (3) and a spot function F2 describing    a regular amplitude-modulation screen. This combination is performed    in the frequency domain. The module A2 and the phase B2 of the    Fourier transform of F2 are calculated. A first matrix A1 is then    generated with a visible pattern according to 1. The position of the    N largest peaks in the matrix A2 is then recorded, and a circular    region centered around the corresponding positions in A1 is set to    zero. A third matrix A3 is calculated as a combination of the two    matrices A1 and A2. This combination can take the form of an    addition (A3=A1+A2), a multiplication (A3=A1*A2), a linear    interpolation (A3=(1−s)*A1+s*A2, with s in ]0, 1[), etc. The two    matrices A3 and B2 are converted to a single matrix of complex    numbers (C), with C(x,y)=A3(x,y)*exp(i*F2(x,y)). C is then made    symmetrical (FFT sense). The spot function used for generating the    security image is obtained by calculating the inverse Fourier    transform of C.-   5. A third class of spot functions is based on some spot function F1    of the first class (3). The dither matrix derived from F1 is applied    to a grayscale image with a constant intensity level. The result of    this operation is a bi-level halftone image B. A set of    morphological operations are applied to B in order to obtain a set    {H₁, H₂, . . . H_(n)} of n bi-level halftones. These morphological    operations may include erosion, dilation, skeletonization, outline,    pruning, among others. The ratio of black pixels {k₁, k₂, . . .    k_(n)} is calculated for each one of the halftones {H₁, H₂ . . .    H_(n)}. These ratios of black pixels {k₁ . . . k_(n)} are associated    to the corresponding halftones. The set of halftones is then ordered    according to these ratios. The individual halftones are merged    together in to order form the spot function F used for generating    the security image. This merging is done by traversing all the    pixels F(x,y) of F. For each pixel, the values {H₁(x,y), H₂(x,y), .    . . H_(n)(x,y)} of the corresponding pixel in {H₁, H₂ . . . H_(n)}    are retrieved. The highest value max_(k)(H_(k)(x,y)) is assigned to    F(x,y). taking the highest value-   6. A fourth class of spot functions are directly derived from some    of the patterns described in 1.4. If the distribution of the 1D    signal used to build a pattern is balanced enough, i.e. the set of    values taken by the 1D signal is evenly distributed, (it takes a    “large enough” set of values) then it may be used directly as a spot    function. This is particularly interesting for LRHFs. Indeed, since    the Fourier transform of a LRHF is also a LRHF, the same detector    may be used.-   7. This particular property enables to combine in the spatial domain    two kind of signals in distinct (or even overlapping) areas:    -   areas featuring security patterns defined by the inverse Fourier        transform of matrix C    -   areas featuring a security pattern defined by C itself        This combination of signal enables for instance to use the        security pattern as an overt decorative image in some areas        (because of its circular symmetries and invariance properties,        the matrix C has some aesthetic properties as can be seen on        FIG. 41 and FIG. 42), or as a covert invisible security in other        areas. This approach can be better understood with FIG. 43. A        banknote 2710 features different areas 2705, 2706, 2707 with        arbitrary size and location which are partly overlapping        (overlap may be obtained by overprint or by digital        combination). Each of these areas is filled with a security        pattern which is obtained by one of the above methods: the area        2705 is obtained by tiling a circular log invariant function,        the area 2706 is obtained by tiling the inverse Fourier        transform of this circular function, the area 2707 is obtained        by tiling the skeletonized and thresholded version of this        inverse Fourier transform. Each individual pattern will        contribute in the Fourier space (modulus image) to increase the        signal to noise ratio of the circular signal. This approach can        be easily generalized with other integral transform than        Fourier.-   8. A fifth class of spot functions are built by combining spot    functions of the four other classes with operations such as    addition, subtraction, multiplication, exclusive-or, addition modulo    n.

In another embodiment, the security image C(x,y)=A(x,y)*exp(i*B(x,y))defined above in the Fourier domain with a rotating 1D function forA(x,y) and a quantum random signal or a pseudo-random signal for B(x,y)is directly printed as an overlay on the banknote to be protected. Forinstance, a banknote is first printed with 4 different ink colors. Thesecurity image (see image 1203 in FIG. 35) layer is afterward overlaidwith a separate color all over the already printed banknote. This colorshould be chosen in order to obtain the best compromise betweeninvisibility and detectability of the signal. For instance, a light graycolor may be an appropriate choice for a banknote featuring little or nographic (like in the water mark area). A darker ink may be required inother cases. Ideally, the color of the security image should be chosenamong the already used set of colors (4 in our example) in order tominimize the number of offset plates.

The main problem that arises when overlaying over a non uniform arealike a banknote is to obtain areas where the security image is eithertoo visible (thus degrading the visual appearance of the banknote) ornot enough visible (thus not reliably detectable). One solution is tolocally increase or decrease the intensity of the security image basedon a weighting function W(x,y) as shown in the first embodiment. Anothersolution consists in adjusting the transparency of the ink used forproducing the signal overlay: a transparent ink will produce a faintlyvisible signal on all but the lightest backgrounds, while an opaque inkwill produce a strongly visible signal on most backgrounds. In anotherembodiment, the security pattern is obtained by combining the fourth andninth embodiment: a banknote includes some areas with a grating andother areas filled with an invariant signal.

The integration of the signal into the design layout of the banknote canbe performed as illustrated in FIG. 49: In 2500, the signal is digitallyinjected into the artwork 2511 (either by modifying the dither matricesor by digital overlay) of the banknote with a strength 2510. In 2501, anestimate of the signal intensity is computed. This estimate is aprediction of what will be the signal intensity after printing andscanning and is compared to some predefined threshold in 2502 (thisthreshold can be the minimum number of lines required for a positivedetection). If intensity is not sufficient, then strength 2510 isincreased and the process repeats. The whole process may be entirelyautomatic (the system automatically adjusts to the minimum strengthrequired for positive detection) or interactive (the designer can thenevaluate the visual impact of a given strength on the design and on thedetectability). This adjustment process may be non-iterative if it ispossible to predict exactly the strength required for a given artwork2511.

1. A method for generating a printed security image on a banknote, saidsecurity image comprising an original image and a security pattern, themethod comprising the steps of: obtaining said security pattern, whereinsaid security pattern in the spatial domain is the inverse Fouriertransform of the combination in the frequency domain between the Fouriertransform of an auxiliary image and a radially symmetric,two-dimensional pattern, said two-dimensional pattern created bysweeping a self-similar, one-dimensional function along a 360-degreearc, said security pattern being detectable from the maximum value ofthe cross-correlation of said one-dimensional function with the Fouriertransform of one line of said banknote, said line being sampled alongany arbitrary direction and with any resolution between 50 and 1200 dotsper inch; generating the security image in the spatial domain by mergingat least one color of at least a part of the original image with thesecurity pattern.
 2. The method as defined in claim 1, wherein themerging of the original image with the security pattern for generatingthe security image is performed by thresholding a grayscale image with aspot function, the module of the Fourier transform of said spot functioncontaining a two-dimensional radially symmetric pattern.
 3. The methodof claim 2, wherein the spot function is constructed as the inverseFourier transform of a module and a phase, said module being a radiallysymmetric pattern, and said phase being white noise.
 4. The method ofclaim 2, wherein the spot function is constructed as the inverse Fouriertransform of a module and a phase, said module constructed as a linearinterpolation between a pair of two dimensional patterns, the firstpattern being the circularly symmetric pattern constructed by sweepingthe self-similar one-dimensional signal along the 360-degree arc and thesecond pattern being the module of the Fourier transform of an arbitraryspot function, and said phase being the phase of the Fourier transformof said arbitrary spot function.
 5. The method of claim 2, wherein themodule of the Fourier transform of the spot function contains atwo-dimensional radially symmetric pattern constructed by sweeping anauto-correlated one-dimensional signal along a radially symmetric curve.6. The method of claim 2, wherein the module of the Fourier transform ofthe spot function contains, a two-dimensional radially symmetric patternconstructed by sweeping a scale-invariant one-dimensional signal along aradially symmetric curve.
 7. The method of claim 1, wherein thetwo-dimensional radially symmetric pattern is also circularly symmetricand constructed by sweeping a one-dimensional signal along a 360 degreearc.
 8. The method of claim 7, wherein the two-dimensional circularlysymmetric pattern is constructed by sweeping an auto-correlatedone-dimensional signal along a 360 degree arc.
 9. The method of claim 7,wherein the two-dimensional circularly symmetric pattern is constructedby sweeping a scale-invariant one-dimensional signal along a 360 degreearc.
 10. The method of claim 1, wherein the module of the Fouriertransform of the security image contains a two-dimensional radiallysymmetric pattern constructed by sweeping an auto-correlatedone-dimensional signal along a radially symmetric curve.
 11. The methodof claim 1, wherein the module of the Fourier transform of the securityimage contains a two-dimensional radially symmetric pattern constructedby sweeping a scale-invariant one-dimensional signal along a radiallysymmetric curve.
 12. The method as defined in claim 1, wherein thesecurity image is concealed.
 13. The method as defined in claim 1,wherein the security image fulfils a decorative function.
 14. The methodas defined in claim 1, wherein multiple security images produced withdifferent sweeping curves are used.
 15. The method as defined in claim1, wherein multiple security images produced with differentone-dimensional signals are used.
 16. The method as defined in claim 1,wherein multiple security images produced with different mergingtechniques are used.
 17. The method as defined in claim 1, wherein themerging technique is an overlay of the security image on the originalimage.
 18. The methods as defined in claim 17, wherein the overlay oftwo images is performed by printing the first image onto the second. 19.The method as defined in claim 1, wherein the merging technique is anoverlay of the original image on the security image.
 20. The method asdefined in claim 1, wherein the security pattern can also be identifiedby combining successive lines.
 21. The method as defined in claim 1,wherein the detection is performed by combining successive lines.
 22. Amethod for generating a printed security image on a banknote, saidsecurity image comprising an original image and a security pattern, themethod comprising the steps of: obtaining said security pattern, whereinsaid security pattern in the spatial domain is an inverse integraltransform of the combination in the frequency domain between an integraltransform of an auxiliary image and a radially symmetric,two-dimensional pattern, said two-dimensional pattern created bysweeping a self-similar, one-dimensional function along a 360-degreearc, said security pattern being detectable from the maximum value ofthe cross-correlation of said one-dimensional function with the integraltransform of one line of said banknote, said line being sampled alongany arbitrary direction and with any resolution between 50 and 1200 dotsper inch; generating the security image in the spatial domain by mergingat least one color of at least a part of the original image with thesecurity pattern.